Inferring a probability density function (pdf) for shape from a population of point sets is a challenging problem. The lack of point-to-point correspondences and the non-linearity of the shape spaces undermine the linear models. Methods based on manifolds model the shape variations naturally, however, statistics are often limited to a single geodesic mean and an arbitrary number of variation modes. We relax the manifold assumption and consider a piece-wise linear form, implementing a mixture of distinctive shape classes. The pdf for point sets is defined hierarchically, modeling a mixture of Probabilistic Principal Component Analyzers (PPCA) in higher dimension. A Variational Bayesian approach is designed for unsupervised learning of the posteriors of point set labels, local variation modes, and point correspondences. By maximizing the model evidence, the numbers of clusters, modes of variations, and points on the mean models are automatically selected. Using the predictive distribution, we project a test shape to the spaces spanned by the local PPCA's. The method is applied to point sets from: i) synthetic data, ii) healthy versus pathological heart morphologies, and iii) lumbar vertebrae. The proposed method selects models with expected numbers of clusters and variation modes, achieving lower generalization-specificity errors compared to state-of-the-art.

Statistical shape models (SSM) are used to introduce shape priors in the segmentation of medical images. However, such models require large training datasets in the case of multi-object structures, since it is required to obtain not only the individual shape variations but also the relative position and orientation among objects. A solution to overcome this limitation is to model each individual shape independently. However, this approach does not take into account the relative position, orientations and shapes among the parts of an articulated object, which may result in unrealistic geometries, such as with object overlaps. In this article, we propose a new Statistical Model, the Statistical Interspace Model (SIM), which provides information about the interaction of all the individual structures by modeling the interspace between them. The SIM is described using relative position vectors between pair of points that belong to different objects that are facing each other. These vectors are divided into their magnitude and direction, each of these groups modeled as independent manifolds. The SIM was included in a segmentation framework that contains an SSM per individual object. This framework was tested using three distinct types of datasets of CT images of the spine. Results show that the SIM completely eliminated the inter-process overlap while improving the segmentation accuracy.

Detailed segmentation of the vertebrae is an important pre-requisite in various applications of image-based spine assessment, surgery and biomechanical modeling. In particular, accurate segmentation of the processes is required for image-guided interventions, for example for optimal placement of bone grafts between the transverse processes. Furthermore, the geometry of the processes is now required in musculoskeletal models due to their interaction with the muscles and ligaments. In this paper, we present a new method for detailed segmentation of both the vertebral bodies and processes based on statistical shape decomposition and conditional models. The proposed technique is specifically developed with the aim to handle the complex geometry of the processes and the large variability between individuals. The key technical novelty in this work is the introduction of a part-based statistical decomposition of the vertebrae, such that the complexity of the subparts is effectively reduced, and model specificity is increased. Subsequently, in order to maintain the statistical and anatomic coherence of the ensemble, conditional models are used to model the statistical inter-relationships between the different subparts. For shape reconstruction and segmentation, a robust model fitting procedure is used to exclude improbable inter-part relationships in the estimation of the shape parameters. Segmentation results based on a dataset of 30 healthy CT scans and a dataset of 10 pathological scans show a point-to-surface error improvement of 20% and 17% respectively, and the potential of the proposed technique for detailed vertebral modeling.